Distance magic labeling of the halved folded n -cube

Hypercube is an important structure for computer networks. The distance plays an important role in its applications. In this paper, we study a magic labeling of the halved folded n -cube which is a variation of the n -cube. This labeling is determined by the distance. Let G be a finite undirected simple connected graph with vertex set V ( G ), distance function ∂ and diameter d . Let D⊆{0,1,… ,d} be a set of distances. A bijection l:V(G)→{1,2,… ,|V(G)|} is called a D -magic labeling of G whenever ∑ _x∈ G_D(v)l(x) is a constant for any vertex v∈ V(G) , where G_D(v)={x∈ V(G): ∂ (x,v)∈ D} . A {1} -magic labeling is also called a distance magic labeling. We show that the halved folded n -cube has a distance magic labeling (resp. a {0,1} -magic labeling) if and only if n=16q^2 (resp. n=16q^2+16q+6 ), where q is a positive integer.